The existence of a solution to the Orlicz chord Minkowski problem is established under a sufficient condition, which is an extension of the L_p chord Minkowski problem for p>1 .

This paper is to generalize convex functions to p -convex functions to dual p -convex functions. Concepts such as p -mean, Legendre transformation, convex base, and p -sum are described for dual p -convex functions. In particular, both the differential formula for dual p -convex functions and the reverse Borell–Brascamp–Lieb inequality are provided. Moreover, a reverse logarithmic Sobolev inequality is also proved by the dual Hamilton–Jacobi equations.

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations.

To handle the problem of not completely incorporating profits of ancillary services and effectively coordinating with the long-time scale fluctuations of wind power, this paper proposes a sizing optimization method for hydrogen energy storage system (HESS) with comprehensive ancillary services and coordination with actual fluctuations of wind power. A model aiming at maximizing comprehensive profit is established including comprehensive ancillary services profit of HESS in electricity market, such as peak regulation, frequency regulation and delayed line construction. The model incorporates the longtime fluctuation characteristics of wind power. The HESS with long-time scale characteristics is more suitable for the actual wind power fluctuations. As a result, a more optimal sizing with larger profits can be obtained. Finally, the particle swarm optimization (PSO) algorithm is applied to solve the nonlinear mixed integer programming model. The numerical simulation results demonstrate that the proposed model can obtain more profits compared to the previous methods on sizing optimization of HESS.

In the context of the "dual-carbon" vision strategy and the development of carbon finance, the development of China's carbon market is very promising, but the system of carbon futures, carbon options and other financial products has not yet been formed. In view of the experience of the European Climate Exchange (ECX) in carbon option contracts, this study aims to design an option product applicable to the Chinese carbon market. The article takes the carbon emission allowances in the national carbon market as the underlying asset of the options, selects the national carbon emission allowance trading price data for the period from July 16, 2021 to December 31, 2022, and uses the EGARCH-fractional Brownian Motion option pricing model to formulate a reasonable call price of the carbon options.

All continuous and SL(n) covariant matrix-valued valuations on functions with finite second moments are completely classified without the symmetry assumption. The moment matrix is shown to be the unique such valuation if n >= 3, while it is essentially the only such valuation in dimension two.

In this paper, we establish a characterization of the polarity mapping for 1-dimensional convex bodies, which is a supplement to the result for such a characterization obtained by Böröczky and Schneider.

We establish a characterization of the star duality mapping for star-shaped sets in n-dimensional Euclidean vector space.

The logarithmic Minkowski problem for q-capacity asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the logarithmic q-capacitary measure of a convex body. This paper solves the case of discrete measures whose supports are in general position.

Some Orlicz-Brunn-Minkowski type inequalities for(dual)quermassintegrals of polar bodies and star dual bodies have been introduced.In this paper,we generalize the results and estab-lish some Orlicz-Brunn-Minkowski type inequalities for mixed(dual)quermassintegrals of polar bodies and star dual bodies.

A complete classification of continuous $$\text {SL}(n)$$ covariant vector-valued valuations on $$L^{p}({\mathbb {R}}^{n},|x|dx)$$ is obtained without any homogeneity assumptions. The moment vector is shown to be essentially the only such valuation.

Sharp complex Lp affine isoperimetric inequalities are established for the entire class of complex Lp projection bodies and the entire class of complex Lp moment bodies.

All $\textrm{SL}(n)$ equivariant matrix-valued valuations on polytopes in $\mathbb{R}^{n}$ are completely classified without any continuity assumptions. Moreover, the symmetry assumption of matrices is removed. The moment matrix is shown to be the only such valuation if $n\geq 4$, while new functionals show up in dimensions two and three.

All $$\mathrm{SL}(n)$$ contravariant vector valuations on polytopes in $${\mathbb {R}}^n$$ are completely classified without any additional assumptions. The facet vector is defined. It turns out to be the unique class of such valuations for $$n\ge 3$$ . In dimension two, the classification corresponds to the known case of $$SL (2)$$ covariant valuations.

This paper is to generalize the dual mixed volume of star bodies to that of dual quasi-concave functions by extending the radial Minkowski linear combination of star bodies to that of dual quasi-concave functions. We attempt to build up some functional versions of notions and inequalities from the dual Brunn-Minkowski theory. In particular, both the dual mixed Brunn-Minkowski inequality of star bodies and the dual Aleksandrov Fenchel inequality of star bodies are generalized to that of dual quasi-concave functions.

All SL(n) contravariant symmetric matrix valued valuations on convex polytopes in R-n are completely classified without any continuity assumptions. The general Lutwak-Yang-Zhang matrix is shown to be essentially the unique such valuation.

All SL(n) contravariant $$L_{p}$$ harmonic valuations on convex polytopes are completely classified without homogeneity assumptions.